Compressed Sensing using Generative Models
This addresses the compressed sensing problem for domains where sparsity is not applicable, offering a novel approach with significant measurement reduction.
The paper tackles the problem of compressed sensing by replacing sparsity assumptions with the assumption that vectors lie near the range of a generative model, achieving an ℓ2/ℓ2 recovery guarantee with roughly O(k log L) random Gaussian measurements. It demonstrates that this method can use 5-10x fewer measurements than Lasso for the same accuracy.
The goal of compressed sensing is to estimate a vector from an underdetermined system of noisy linear measurements, by making use of prior knowledge on the structure of vectors in the relevant domain. For almost all results in this literature, the structure is represented by sparsity in a well-chosen basis. We show how to achieve guarantees similar to standard compressed sensing but without employing sparsity at all. Instead, we suppose that vectors lie near the range of a generative model $G: \mathbb{R}^k \to \mathbb{R}^n$. Our main theorem is that, if $G$ is $L$-Lipschitz, then roughly $O(k \log L)$ random Gaussian measurements suffice for an $\ell_2/\ell_2$ recovery guarantee. We demonstrate our results using generative models from published variational autoencoder and generative adversarial networks. Our method can use $5$-$10$x fewer measurements than Lasso for the same accuracy.